Classification and Decomposition Module

• The paper introduces a rigorous characterization of lattice decompositions via central, fully invariant idempotent endomorphisms in module theory.
• The criteria in commutative cases leverage support sets and distributive submodules to ensure that decompositions lift to injective hulls.
• Examples and categorical equivalences in σ[M] clarify practical applications and limitations, particularly in noncommutative settings.

A classification and decomposition module, in the setting of module theory and lattice theory, refers to the study of when the submodule lattice L(M) of a module M (over a ring R) decomposes as a direct product of lattices, and how such decompositions are classified, constructed, and related to other categorical structures. The topic provides a rigorous characterization via idempotent endomorphisms, a transparent criterion in the commutative case via module support, and a linkage to the Grothendieck subcategory σ[M]. The following sections systematically develop the foundational aspects, main theorems, special cases, categorical frameworks, and illustrative examples establishing the scope and power of the classification and decomposition principle for module sublattices (García et al., 2021).

1. Foundational Definitions and Structures

Let M be a right module over a ring R. The submodule lattice L(M) is the set of all submodules X ≤ M, equipped with meet (intersection) and join (sum). L(M) is a modular, bounded lattice, serving as a fundamental invariant in module theory.

A lattice decomposition of M is a direct-sum decomposition M = N ⊕ H such that the submodule lattice L(M) is isomorphic to the product lattice L(N) × L(H). This is a strict refinement of the well-known direct-sum decomposition and imposes additional compatibility conditions at the level of submodules.

The endomorphism ring S = End_R(M) plays a central role. A decomposition M = N ⊕ H is always associated with an idempotent e ∈ S satisfying e(M) = N, (1 − e)(M) = H. However, not all such idempotents induce a lattice decomposition; further invariance and centrality properties are required.

2. Idempotent Endomorphisms and Lattice Decomposition

The central results establish necessary and sufficient conditions under which an idempotent endomorphism e ∈ S specifies a lattice decomposition of M. The main theorem asserts:

• M = N ⊕ H gives rise to a lattice decomposition L(M) ≅ L(N) × L(H) if and only if:
  1. e is a central idempotent in S (i.e., e commutes with all of S),
  2. e is fully invariant: for any submodule X ≤ M, e(X) ≤ X,
  3. N and H are distributive elements in the lattice L(M).

The distributivity condition on N (and similarly H) is characterized by: for all submodules Y₁, Y₂ ≤ M,

$(Y₁ ∧ Y₂) + N = (Y₁ + N) ∧ (Y₂ + N).$

An equivalent module-theoretic criterion is that N and H have no isomorphic simple subfactors and, for all n ∈ N, h ∈ H, Ann_R(n) + Ann_R(h) = R.

In the case where M = R as a right module over itself, lattice decompositions correspond precisely to central idempotents in R, as expected from classical ring theory; R ≅ Re × R(1–e).

3. Criteria in the Commutative Case: Support and Finiteness

For commutative rings R = A, the criterion for lattice decomposition becomes transparent in terms of the support of modules:

• For M = N ⊕ H over A, the following are equivalent:
  • M = N ⊕ H is a lattice decomposition,
  • Support sets satisfy Supp(N) ∩ Supp(H) = ∅,
  • N and H are distributive submodules.

Support is defined as Supp(M) = { p ∈ Spec(A) : M_p ≠ 0 }.

When A is noetherian and M decomposes as above, the decomposition lifts to the injective hull: E(M) = E(N) ⊕ E(H), preserving the distributive structure.

A complemented distributive summand N ⊆ M is precisely one for which there exists an idempotent in the finite-topology closure of A/Ann(M) acting on M as the projection onto N. For finitely generated modules, such N are of the form N = aM for a unique idempotent a ∈ A/Ann(M).

Example: For A = ℤ, M = ℤ/6ℤ, the nontrivial idempotents [3], [4] ∈ ℤ/6ℤ/A/Ann(M) yield the decomposition M = 2M ⊕ 3M, with 2M ≅ ℤ/3ℤ, 3M ≅ ℤ/2ℤ.

4. Noncommutative Existence and Limiting Cases

Beyond the commutative context, the situation is subtler. Theorem 2.2 yields the complete criterion: M has a nontrivial lattice decomposition if and only if S = End_R(M) admits a nontrivial central, fully invariant idempotent.

Concretely, this requires a direct sum decomposition of S, i.e., S = I₁ ⊕ I₂ with Iᵢ two-sided ideals generated by central idempotents e₁, e₂, so M = e₁M ⊕ e₂M.

Examples illustrate the nuance:

• For R = M₂(K) (matrix ring over a field), M = R as right module admits no lattice decomposition since End_R(R) = R has no nontrivial central idempotents.
• For M = ℤ{(2)} ⊕ ℤ{(3)}, although the endomorphism ring is a direct product, the summands are not distributive since the structure of submodules and endomorphism invariance requirements are not fulfilled, precluding lattice decomposition.

5. Decomposition and the Grothendieck Subcategory σ[M]

A robust categorical perspective emerges from the Grothendieck subcategory σ[M], which is the smallest full Grothendieck subcategory of Mod–R containing M. Objects in σ[M] are submodules of quotients of direct sums of copies of M.

Key results:

• If M = N ⊕ H is a lattice decomposition, then for any index set I, M⁽ᴵ⁾ = N⁽ᴵ⁾ ⊕ H⁽ᴵ⁾ is a lattice decomposition.
• The functorial equivalence σ[M] ≅ σ[N] × σ[H] holds if and only if M admits a lattice decomposition. Any X in σ[M] can be decomposed as X = (X ∩ N⁽ᴵ⁾) ⊕ (X ∩ H⁽ᴵ⁾), reflecting a product structure at the categorical level.

For R a finite-dimensional algebra with orthogonal central idempotents e₁, e₂, the decomposition R = e₁R ⊕ e₂R induces the categorical equivalence Mod–R ≅ Mod–e₁R × Mod–e₂R.

6. Illustrative Examples and Limitations

Case studies clarify the general principles:

• M = ℤ/2 ⊕ ℤ/2 is semisimple with length 2. Although End(M) = M₂(ℤ/2) has idempotents, none are central. Thus, no lattice decomposition exists.
• M = ℤ/2 ⊕ ℤ/3, with End(M) = ℤ/2 × ℤ/3, yields two central idempotents (1,0), (0,1), allowing the decomposition ℤ/2 ⊕ ℤ/3 as a lattice decomposition, since their supports are disjoint.
• The injective hull may fail to preserve distributivity in the noncommutative case: for R = K ⊕ K, with K a field, certain module decompositions do not lift to the injective hull in a distributive way, highlighting the limitations outside the commutative case.

7. Summary Table: Existence Criteria

Setting	Lattice Decomposition Exists	Criterion
Commutative (A)	Yes iff Supp(N) ∩ Supp(H) = ∅	Idempotent a ∈ A/Ann(M) acting as projection
Noncommutative (R)	Yes iff central, fully invariant e	S = End_R(M) admits central, fully invariant idempotent e
End(R) = matrix ring	No	No nontrivial central idempotents
σM	Yes iff M = N ⊕ H lattice decomp.	σ[M] ≅ σ[N] × σ[H] if and only if lattice decomposition exists

These results equip researchers with explicit, structural, and categorical tools for understanding when the submodule lattice of a module splits as a product, the role of endomorphism invariance, and the interplay with module support and Grothendieck subcategories (García et al., 2021).